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\title{The MVT}
\date{April 6, 2022}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\prop{
Let $f: [a,b] \to \RR$ be differentiable at $c \in (a,b)$.
If $f$ has a relative minimum or a relative maximum at $c$, then $f'(c) = 0$.
}

\thm{[Rolle's theorem]
Let $f: [a,b] \to \RR$ be continuous with $f(a) = f(b)$.
If $f$ is differentiable on $(a,b)$, then there is $c \in (a,b)$ such that $f'(c) = 0$.
}

\prob{
Prove Rolle's theorem.
}

\vfill

\prob{
Find an example of a continuous function $f:[0,1] \to \RR$ such that $f(0)=f(1)$ but $f'(c) \neq 0$ for every $c \in (0,1)$ at which $f$ is differentiable.
Try to minimize the number of points at which $f$ isn't differentiable.
}

\vspace{2in}

\newpage

\thm{[Mean value theorem]
Let $f:[a,b] \to \RR$ be continuous and also differentiable on $(a,b)$.
Then there is $c \in (a,b)$ such that $f(b) - f(a) = f'(c)(b-a)$. 
}

\prob{
Your goal is to prove the MVT.
\begin{enumerate}[a)]
\item Find an equation $y = L(x)$ for the line through the points $(a, f(a))$ and $(b, f(b))$.
\item Define a new function $g:[a,b] \to \RR$ by $g(x) = f(x) - L(x)$.
\item Verify that $g$ is continuous on $[a,b]$ and differentiable on $(a,b)$.
\item Use Rolle's theorem and finish the proof.
\end{enumerate} 
}

\vfill

\cor{
Let $I$ be an interval and let $f: I \to \RR$ be differentiable.
\begin{enumerate}[i)]
\item $f$ is constant if and only if $f'(x) = 0$ for all $x \in I$.
\item $f$ is increasing if and only if $f'(x) \geq 0$ for all $x \in I$.
\item $f$ is decreasing if and only if $f'(x) \leq 0$ for all $x \in I$.
\end{enumerate}
}

\prob{
Prove the corollary.
}

\vspace{1in}

\centering{\tiny(you'll certainly need another sheet of paper)}

\end{document}